Superposition of Waves - Stationary Waves in String

When two waves of the same frequency and amplitude travel in opposite directions along the same medium, they interfere with each other. This interference can be constructive or destructive, depending on the phase difference between the waves. The principle that describes this phenomenon is known as the superposition principle.

Superposition Principle

The superposition principle states that when two or more waves meet at a point, the resultant displacement at that point is equal to the vector sum of the displacements of the individual waves.

Mathematically, if $y_1$ and $y_2$ are the displacements of two waves at a point, the resultant displacement $y$ is given by:

$$ y = y_1 + y_2 $$

Stationary Waves (Standing Waves)

Stationary waves, also known as standing waves, are formed by the superposition of two waves with the same frequency and amplitude traveling in opposite directions. Unlike traveling waves, stationary waves do not propagate through the medium. Instead, they exhibit nodes and antinodes.

• Nodes: Points of zero amplitude where destructive interference occurs.
• Antinodes: Points of maximum amplitude where constructive interference occurs.

Formation of Stationary Waves in a String

When a string fixed at both ends is plucked or otherwise disturbed, waves travel along the string, reflect off the fixed ends, and interfere with the incoming waves. This can result in the formation of stationary waves if the frequency of the disturbance is at or near one of the string's natural frequencies (harmonics).

Conditions for Stationary Waves

For a stationary wave to form on a string, the length of the string $L$ must be an integral multiple of half the wavelength $\lambda$ of the wave:

$$ L = n \frac{\lambda}{2} \quad \text{where} \quad n = 1, 2, 3, \ldots $$

The frequencies at which stationary waves form are called the resonant frequencies or harmonics of the string.

Frequency of Stationary Waves

The frequency $f$ of the $n$-th harmonic of a string of length $L$, tension $T$, and linear mass density $\mu$ (mass per unit length) is given by:

$$ f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}} $$

Differences Between Stationary Waves and Traveling Waves

Aspect	Stationary Waves	Traveling Waves
Energy Propagation	Energy does not propagate.	Energy propagates through medium.
Displacement	Nodes have zero displacement.	Displacement varies continuously.
Amplitude	Amplitude varies along the medium.	Amplitude is constant.
Phase	Phase is constant between nodes.	Phase changes with propagation.
Formation	Formed by interference.	Formed by a single disturbance.

Examples

Example 1: Fundamental Frequency

Consider a string fixed at both ends with a length of 1 meter, tension of 100 N, and a mass per unit length of 0.01 kg/m. What is the fundamental frequency (first harmonic) of the string?

Using the formula for the frequency of stationary waves:

$$ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} $$

Substituting the given values:

$$ f_1 = \frac{1}{2 \times 1} \sqrt{\frac{100}{0.01}} = \frac{1}{2} \times 100 = 50 \text{ Hz} $$

Example 2: Third Harmonic

Using the same string as in Example 1, what is the frequency of the third harmonic?

$$ f_3 = \frac{3}{2L} \sqrt{\frac{T}{\mu}} = 3 \times f_1 = 3 \times 50 = 150 \text{ Hz} $$

Conclusion

Understanding the superposition of waves and the formation of stationary waves in a string is crucial for various applications in physics, engineering, and music. The ability to predict and manipulate the resonant frequencies of a string allows for the design of musical instruments, the study of acoustics, and the exploration of wave phenomena in other contexts.